All of the 3rd grade teachers and students from Santa Rita went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$3.50$ each for students, and the group paid $$52.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$34.00$ each for teachers and $$12.00$ each for students, and the group paid $$188.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+3.5y = 52}$ ${34x+12y = 188}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-14y = -208}$ ${34x+12y = 188}$ Add the top and bottom equations together. $ -2y = -20 $ $ y = \dfrac{-20}{-2}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {8.5x+3.5y = 52}$ to find $x$ ${8.5x + 3.5}{(10)}{= 52}$ $8.5x+35 = 52$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {34x+12y = 188}$ and get the same answer for $x$ ${34x + 12}{(10)}{= 188}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.